# Elementary dynamics of particles and solids

ELEMENTARY DYNAMICS OF PARTICLES AND SOLIDS

BY W. M. HICKS,

LONDON, MACMILLAN AND CO., 1890

Elementary dynamics of particles and solids

PREFACE

The following treatise is based on courses of lectures delivered in Firth College, Sheffield. It is intended as an introduction to the principles of dynamics for the use of students with no knowledge of mathematics beyond the elements of algebra and pure geometry. It will thus be found useful, not only in colleges and schools, but also to that large class of mechanical engineers to whom a knowledge of dynamics is valuable, but whose acquaintance with mathematics is slight. The wants of this latter class have been kept in view throughout, although developments properly to be found in technical treatises are not introduced. The chapters on the motion of rigid bodies will, it is hoped, be especially useful to them. This part will also be of value to ordinary students, who will thus be introduced to the simple principles of rigid dynamics freed from the intricacies of the differential and integral calculus, which usually accompany them. For engineers, the knowledge of the properties of moments of inertia and their values for simple bodies are as important as that of their centres of gravity, and it is hoped that the methods of Chapter XIX will be found as simple as those employed in finding centres of gravity.

Although a knowledge of trigonometry has not been assumed in the text, it has been occasionally introduced in some of the worked-out examples, and examples have been added which require trigonometry for their solution. The book is thus rendered useful to a larger circle of students.

The chief points of novelty in the presentment of the subject are the following: (1) no separation has been made between Statics and Kinetics - but they have been considered together, the former merely as a special case of the latter; (2) the way in which the idea of Mass and its measure is introduced, and the discussion of Momentum before that of Force, depart from the order usually followed. The author believes, however, that this is the only logical way of treating the subject, based, as it must be ultimately, on experimental laws. This gives to the student a vivid realisation of the essential property of matter and inertia at the very commencement of the subject. He would strongly recommend the student himself to perform or see the experiments described.

CONTENTS

Introduction

PART I: RECTILINEAR MOTION OF A PARTICLE
1. Motion in a Straight Line
2. Mass - Momentum - Collision
3. Force
4. Work - Energy - Power
5. Units

PART II: FORCES IN ONE PLANE
6. Composition of Velocities and Accelerations
7. Coplanar Forces acting at one Point
8. Forces on a Rigid Body - Parallel Forces
9. Forces on a Rigid Body - Centre of Gravity
10. Miscellaneous theorems and examples
11. Energy
12. Machines
13. Friction
14. Oblique Impact
15. Projectiles
16. Motion on Curves
17. Simple Harmonic Motion - Pendulum

PART III: MOTION OF RIGID BODIES
18. Kinematics of Motion of Solids
19. Momentum and Energy
20. Moments of Inertia
21. Motion under Force

Miscellaneous Examples

CHAPTER II: MASS - MOMENTUM - COLLISION

In this chapter we enter on the consideration of matter in its relations to motion. The quantity of matter in a given body is called its mass. To measure this mass we must refer it to another portion of matter which we take for unit mass, and say how many times it contains this unit; and this leads us to the question how this ratio is to be determined.

Now, in general, in order to determine the ratio between two quantities of the same kind, the first step is to get some criterion which shall determine when they are equal. For instance, suppose we have a given thing A (whether mass or any other quantity) and a criterion by which to determine if it is equal to another thing B of the same kind. It will then be possible, by applying the criterion of equality, to make any number of things of the same kind each equal to A. By combining two of them we can get one whose magnitude is twice that of A; by combining three one whose magnitude is three times that of A, and so on. In a similar way we can obtain two equal ones which together equal A - in other words, we can get one whose magnitude is one-half that of A, and so on. Having obtained these larger and smaller units we can, by combining them, build up a quantity whose magnitude (according to the criterion) is equal to that of B which we wished to measure. To illustrate this, consider the case of measurement of length, the unit A being one foot - i.e. the length of a certain rod. Here the "criterion of equality" is, that when laid alongside with one end coinciding, the other coincides also.

Applying this criterion we can make a series of rods each equal in length to one foot. So also, by making twelve equal pieces which together equal the one foot, we get other units equal to one-twelfth of a foot. If then we wish to measure the length of anything, say the side of a table, we first lay along it end to end a series of our foot rods, until one more would be too long; then a series of the smaller, until one more would be too long, and so on. We have then built up a length equal to that whose measure is required, and the ratio of this composite one to the unit - one foot - is known by the process of its formation. In practice this is simplified, but the rationale is as here given, and is the same whatever the kind of quantity we may wish to measure.

The first point then to be determined is the criterion of equality of two masses, and this criterion must depend on the properties of matter. For a knowledge of these properties recourse must be had to experiment.

In general, when two bodies impinge they fly apart after the collision, but if by any means this is prevented, such as by a catch or sticky cement, the single body composed of these two will in general still have a motion different from that of either before impact. Now suppose two particles to impinge on one another in opposite directions in the same straight line with equal speeds, and to stick to one another. If they do not come to rest, clearly the masses are not equal, for everything else is the same for both. If they do come to rest we shall say that they are equal. This will give the necessary criterion for equal masses; it may be defined as follows.

Two masses are equal, if when they are made to impinge on one another in opposite directions with equal speeds, and stick together, they come to rest. It is supposed here that no rotations are set up.

A simpler practical method of determining equality of masses will be deduced later based on another property of matter. The above is, however, the simplest and most fundamental conception of equal masses.

The criterion supposes that the actual value of the relative velocity of the masses has no influence. Before therefore applying it, it will be necessary to see by experiment whether two masses which, according to the criterion, are equal with one velocity, are also equal with another velocity. To test this and also to employ our definition in the measurement of actual masses it is necessary to have some instrument by which it is possible to give determinate velocities to the bodies to be compared, and to measure their velocities after impact. Such an instrument may be called a ballistic balance. One form of construction is as follows.